A total of 150 centimeters of wire is used to form two separate shapes: a square and an equilateral triangle....

1. TRANSLATE the problem information

• Given information:
  • Total wire used: 150 cm
  • Equation: \(\mathrm{4s + 3t = 150}\)
  • \(\mathrm{s}\) = side length of square = 15 cm
  • \(\mathrm{t}\) = side length of triangle (unknown)

• What this tells us: The \(\mathrm{4s}\) represents the perimeter of the square (4 sides × s), and \(\mathrm{3t}\) represents the perimeter of the equilateral triangle (3 sides × t). Together they use exactly 150 cm of wire.

2. INFER the solution approach

• Since we know \(\mathrm{s = 15}\), we can substitute this value directly into the equation
• This will give us a simple linear equation with only one unknown (t)

3. SIMPLIFY through substitution

• Substitute \(\mathrm{s = 15}\) into \(\mathrm{4s + 3t = 150}\):
  \(\mathrm{4(15) + 3t = 150}\)

• Calculate: \(\mathrm{4 \times 15 = 60}\)
  \(\mathrm{60 + 3t = 150}\)

4. SIMPLIFY to isolate the variable

• Subtract 60 from both sides:
  \(\mathrm{3t = 150 - 60}\)
  \(\mathrm{3t = 90}\)

• Divide both sides by 3:
  \(\mathrm{t = 90 \div 3 = 30}\)

Answer: 30 cm

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not understand what the equation \(\mathrm{4s + 3t = 150}\) actually represents geometrically. They might see it as abstract algebra without connecting it to the wire constraint.

This confusion can lead them to misinterpret the problem or attempt incorrect approaches, causing them to get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors during the substitution process, such as calculating \(\mathrm{4(15)}\) incorrectly as 40 instead of 60, or making errors when subtracting or dividing.

These calculation mistakes typically lead them to select Choice (A) (15) or Choice (C) (90) instead of the correct answer.

The Bottom Line:

This problem tests whether students can connect algebraic equations to real-world geometric constraints and then execute accurate algebraic manipulation. The key insight is recognizing that perimeter formulas create the constraint equation, and substitution provides a direct path to the solution.